.TH std::assoc_laguerre,std::assoc_laguerref,std::assoc_laguerrel 3 "2024.06.10" "http://cppreference.com" "C++ Standard Libary"
.SH NAME
std::assoc_laguerre,std::assoc_laguerref,std::assoc_laguerrel \- std::assoc_laguerre,std::assoc_laguerref,std::assoc_laguerrel

.SH Synopsis
   double      assoc_laguerre ( unsigned int n, unsigned int m, double x );

   double      assoc_laguerre ( unsigned int n, unsigned int m, float x );
   double      assoc_laguerre ( unsigned int n, unsigned int m, long double x );  \fB(1)\fP
   float       assoc_laguerref( unsigned int n, unsigned int m, float x );

   long double assoc_laguerrel( unsigned int n, unsigned int m, long double x );
   double      assoc_laguerre ( unsigned int n, unsigned int m, IntegralType x ); \fB(2)\fP

   1) Computes the associated Laguerre polynomials of the degree n, order m, and
   argument x.
   2) A set of overloads or a function template accepting an argument of any integral
   type. Equivalent to \fB(1)\fP after casting the argument to double.

   As all special functions, assoc_laguerre is only guaranteed to be available in
   <cmath> if __STDCPP_MATH_SPEC_FUNCS__ is defined by the implementation to a value at
   least 201003L and if the user defines __STDCPP_WANT_MATH_SPEC_FUNCS__ before
   including any standard library headers.

.SH Parameters

   n - the degree of the polynomial, a value of unsigned integer type
   m - the order of the polynomial, a value of unsigned integer type
   x - the argument, a value of a floating-point or integral type

.SH Return value

   If no errors occur, value of the associated Laguerre polynomial of x, that is (-1)m

   dm
   dxm

   L
   n + m(x), is returned (where L
   n + m(x) is the unassociated Laguerre polynomial, std::laguerre(n + m, x)).

.SH Error handling

   Errors may be reported as specified in math_errhandling.

     * If the argument is NaN, NaN is returned and domain error is not reported.
     * If x is negative, a domain error may occur.
     * If n or m is greater or equal to 128, the behavior is implementation-defined.

.SH Notes

   Implementations that do not support TR 29124 but support TR 19768, provide this
   function in the header tr1/cmath and namespace std::tr1.

   An implementation of this function is also available in boost.math.

   The associated Laguerre polynomials are the polynomial solutions of the equation
   xy,,
   + (m + 1 - x)y,
   + ny = 0.

   The first few are:

     * assoc_laguerre(0, m, x) = 1.
     * assoc_laguerre(1, m, x) = -x + m + 1.
     * assoc_laguerre(2, m, x) =

       1
       2

       [x2
       - 2(m + 2)x + (m + 1)(m + 2)].
     * assoc_laguerre(3, m, x) =

       1
       6

       [-x3
       - 3(m + 3)x2
       - 3(m + 2)(m + 3)x + (m + 1)(m + 2)(m + 3)].

.SH Example


// Run this code

 #define __STDCPP_WANT_MATH_SPEC_FUNCS__ 1
 #include <cmath>
 #include <iostream>

 double L1(unsigned m, double x)
 {
     return -x + m + 1;
 }

 double L2(unsigned m, double x)
 {
     return 0.5 * (x * x - 2 * (m + 2) * x + (m + 1) * (m + 2));
 }

 int main()
 {
     // spot-checks
     std::cout << std::assoc_laguerre(1, 10, 0.5) << '=' << L1(10, 0.5) << '\\n'
               << std::assoc_laguerre(2, 10, 0.5) << '=' << L2(10, 0.5) << '\\n';
 }

.SH Output:

 10.5=10.5
 60.125=60.125

.SH See also

   laguerre  Laguerre polynomials
   laguerref \fI(function)\fP
   laguerrel

.SH External links

   Weisstein, Eric W. "Associated Laguerre Polynomial." From MathWorld — A Wolfram Web
   Resource.
